Pictures of the final time aren't proof enough for pokemon runs, however you could just point a video camera/webcam at your 3ds while you do the run. As long as the quality is good enough to be able to see what you're doing, it will be fine.

A screenshot of the game time doesn't provide enough information - for instance, it doesn't prove that you did it the run in a single segment.

Capture devices for 3DS are available from Loopy via http://3dscapture.com/ or http://www.merki.net/ if you're in Europe, or from Katsukity via http://www.3dsvideocapture.com/

I would recommend Loopy just because some of Katsukity's can suffer from reliability issues, but plenty of people have had no issue with devices from Katsukity.

Im also curious about the sound issue. Can I film the gameplay on my phone and sound somewhere else? if so how can I do that? Also do you have to do it in one sitting or can you save get off then continue from the save point later?